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Q.If the supply function is p=4−5x+x2p = 4 - 5x + x^2, then find the producer's surplus when price is 18.

CBSECBSE Class XII Board 2025Subjective· 5mImportance★★★★★
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At p=18p=18 the supply curve p=x2−5x+4p = x^2 - 5x + 4 gives x0=7x_0 = 7; producer surplus =18(7)−∫07(x2−5x+4) dx=126−1196=6376≈106.17= 18(7) - \int_0^7 (x^2-5x+4)\,dx = 126 - \tfrac{119}{6} = \tfrac{637}{6} \approx 106.17.

Producer's Surplus =p0 x0−∫0x0p dx= p_0\,x_0 - \displaystyle\int_0^{x_0} p\,dx, where p0p_0 = market price, x0x_0 = quantity supplied at p0p_0, and p=f(x)p = f(x) is the supply function.

  1. Supply function: p=4−5x+x2=x2−5x+4p = 4 - 5x + x^2 = x^2 - 5x + 4. Set p0=18p_0 = 18: 18=x2−5x+4⇒x2−5x−14=018 = x^2 - 5x + 4 \Rightarrow x^2 - 5x - 14 = 0.
  2. Factor: (x−7)(x+2)=0⇒x=7(x-7)(x+2) = 0 \Rightarrow x = 7 (take x0=7x_0 = 7; x=−2x = -2 rejected as quantity cannot be negative).
  3. Revenue term: p0x0=18×7=126p_0 x_0 = 18 \times 7 = 126. …

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